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<title>Atlas software user guide -- Blocks</title>
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<h2>Blocks</h2>
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<i>Last updated: October 16, 2005</i>
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Let G be a connected reductive complex group, with a fixed
<a href="realforms.html">involution</a> &theta;; let K be the group of 
&theta;-fixed points in G, and <b>g</b>=Lie(G).
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Fix a regular integral infinitesimal character &chi; for G. The category of 
(finite-length)
(<b>g</b>,K)-modules with generalized infinitesimal character &chi; naturally
splits up as a finite direct sum of "indecomposables"; the summands are called 
the <i>blocks</i> of the category. A definition is as follows: consider the
set of (classes of) irreducible (<b>g</b>,K)-modules with infinitesimal
character &chi;, and put an edge between two irreducibles when there is a
non-trivial Ext<sup>1</sup> between them. Then the blocks correspond to the
connected components of this graph; a module belongs to a block, if all its
irreducible subquotients are in the corresponding connected component, and each
(<b>g</b>,K)-module with generalized infinitesimal character &chi; splits
up canonically as a direct sum of submodules lying in the various blocks.
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<p>
As explained <a href="parameters.html">here</a>, there is a nice description of
the blocks in terms of the parameters we are using to describe
representations. Each block is again partitioned over the set of conjugacy 
classes of <a href="cartan.html">Cartan subgroups</a> of (G,&theta;); and for 
each Cartan, the resulting piece is an orbit of the natural action of W on the
parameter space. This makes it rather easy to count the number of elements
in the blocks, without having to fully construct the parameter space. The 
&#8220;blocksizes&#8221command will print out the blocksizes as a matrix, where
the rows are indexed by real forms of G, and the columns by real forms
of G<sup>&or;</sup>.
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<i>Example:</i> <b>SL</b>(2). This is much too simple, but it will
at least give an idea. For <b>SL</b>(2), up to equivalence
there are three strong real forms:
two compact ones corresponding to <b>SU</b>(2), and the split one 
<b>SL</b>(2,<b>R</b>). The dual group is <b>PSL</b>(2), and has two strong
real forms, <b>PSU</b>(2) and <b>PSL</b>(2,<b>R</b>).
For the compact real form, picking any of the two corresponding strong forms,
there is a single parameter, corresponding to the split real form of 
<b>PSL</b>(2). For the split real form, there are four parameters, two for
the compact Cartan, and two for the split Cartan. The two for the compact
Cartan correspond to the split real form of <b>PSL</b>(2), because that is
the only real form of the dual group that &#8220;lives&#8221 over that
Cartan. The two parameters for the split Cartan correspond one to the split
dual real form, and one to the compact one. So for <b>SU</b>(2) there is of
course a single block of representations, consisting of a single element,
whereas for <b>SL</b>(2,<b>R</b>) there are two: if we look, say, at the
infinitesimal character of the trivial representation then the one-element
block is the non-spherical principal series, the other is made up of the
trivial representation (lying over the split torus), and the two discrete
series (lying over the compact torus.) This would be the 
&#8220;(split,split)&#8221 block; the two one-element blocks are the
&#8220;(split,compact)&#8221 for <b>SL</b>(2,<b>R</b>) and
&#8220;(compact,split)&#8221 for <b>SU</b>(2), while the 
&#8220;(compact,compact)&#8221 block is empty. Notice that if we looked at an
infinitesimal character where the finite-dimensional representation has even
dimension, then it would be the <i>spherical</i> principal series that is
in a block by itself, the equivalence of categories does not preserve the
action of the center of the group.
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<p>
<i>Example:</i> <b>PSL</b>(2). If we look now at <b>PSL</b>(2), there 
are two translation classes of infinitesimal characters to consider. One is 
dual to the picture obtained for the two compact strong real forms of 
<b>SL</b>(2), and will yield two one-element blocks
(two irreducible principal series of <b>PSL</b>(2,<b>R</b>)); the other 
infinitesimal character will correspond to the picture dual to the one for 
<b>SL</b>(2,<b>R</b>): there will be two blocks, one one-element block for the 
compact real form <b>PSU</b>(2), and a three-element block containing the
one discrete series representation and two finite-dimensional ones for
<b>PSL</b>(2,<b>R</b>)).
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